A discussion of basic principles of Fourier-domain holography is first presented, followed by a discussion of digital holography.
Fourier-Domain Holography
The optical configuration for Fourier-domain digital holographic imaging is shown schematically in FIG. 1, although not to scale. The object plane (x, y) is conjugate to the Fourier plane that is near the output face of the cube beam splitter. The CCD chip is on the Fourier plane. The focal distance between the lens and the Fourier plane is adjusted for the optical path through the glass beamsplitter. The reference plane wave is directed by the beam splitter to intersect with the object wave with a crossing angle θ. The reference wave is incident off-axis, providing a spatial heterodyne signal that modulates the speckle pattern from the object. The interference pattern is recorded on the CCD chip that resides on the Fourier plane (FP). Numerical reconstruction of the image using an FFT is represented as the read-out lens transforming the field back to the space-domain (η, ξ).
An example of a Fourier-domain hologram is shown in FIG. 2(a), with a magnified view of one section shown in FIG. 2(b). This hologram is of lettering on diffusing white paper. The diffuse nature of the target (light scattered into wide angles) ensures that there is a wide recording range on the Fourier plane. The fringe patterns are visible in FIG. 2(b), with a clear periodicity modulated by amplitude and phase across the speckles. The spatial interference fringes modulate the speckle pattern with approximately 2-3 fringes within a speckle coherence length.
The 1-D Fourier transform of the section denoted by the dashed line in FIG. 2(a) is shown in FIG. 3. The zero-order diffraction is the wide base at the center of the graph, including a DC spike at zero spatial frequency. The two broad sidebands at opposite symmetric spatial frequencies are the image information. The spatial frequency width of these first-order peaks is:
                              k          max                =                  k          ⁢                      D                          2              ⁢              f                                                          (        1        )            which is determined by the numerical aperture (f/#) of the imaging optics. A 2-D Fourier transform is shown in FIG. 4 using an Air Force test chart as the target. The power spectrum reconstruction produces an image and it's conjugate. The demodulated and transformed images shown in FIG. 4(a) are the direct image and it's conjugate. FIG. 4(b) is a magnified version of the lower-left reconstruction. Phase information can be retrieved by comparing the real and imaginary parts of the image and it's conjugate.
Digital Holography
Digital holograms (containing N×N=800×800 pixels) are encoded on a CCD chip with 4096 gray levels (12 bits). The pixel size is Δx′=Δy′=6.8 μm and the area of the CCD chip is L×L=5.44×5.44 mm2. The FFT reconstruction of the digital hologram produces an image with N×N pixels with a pixel size Δξ (Δξ=Δη) given by
                    Δξ        =                  Δη          =                                    λ              ⁢                                                          ⁢              f              ⁢                                                          ⁢              Δ              ⁢                                                          ⁢                              v                                  x                  ′                                                      =                                          λ                ⁢                                                                  ⁢                f                ⁢                                                                  ⁢                Δ                ⁢                                                                  ⁢                                  v                                      y                    ′                                                              =                                                λ                  ⁢                                                                          ⁢                  f                                L                                                                        (        2        )            where Δνx′ (Δνx′=Δνy′=1/L) is the sampling spatial frequency.
To record interference fringes in the digital hologram, the fringe spacing should range from twice the pixel size (minimum) to the CCD chip size (in-line holography). The spatial frequency corresponding to the maximum fringe spacing is Δνx′=1/L, and the spatial frequency for the minimum fringe spacing is 1/(2Δx′)=NΔνx′/2, which is the spatial frequency limit. Four times the pixel size (4Δx′) is the best fringe spacing, at which the sideband is located at half of the spatial frequency limit. When the fringe spacing is 4Δx′, the maximum field of view for the holographic image is achieved with NΔξ/2=λf/(2Δx′). The fringe spacing for FIG. 2 was 3Δx′ with the center of the sideband located at the spatial frequency of 1/(3Δx′)=49 mm−1 in FIG. 3.
The transverse resolution in FD-DHOCI (Fourier Domain-Digital Holographic Optical Coherence Imaging) depends on the area of the CCD chip. If the object beam at the Fourier plane covers the full span of the CCD, the transverse resolution at the Rayleigh criterion isRs=1.22λf/L=1.22Δξ.  (3)
The longitudinal resolution depends on the coherence length of the short-coherence source and is
                              Δ          ⁢                                          ⁢          z                =                              ln            ⁡                          (              2              )                                ⁢                      2            π                    ⁢                                                    λ                2                            Δλ                        .                                              (        4        )            where Δλ is wavelength bandwidth of the source intensity coherence envelope. The 12-bit CCD camera has Δx′=6.8 μm, N=800, λ=840 nm, and f=4.8 cm, and the bandwidth of the source is 17 nm. The transverse and the longitudinal resolution for this system are 9 μm and 18 μm, respectively.
In FD-DHOCI, the CCD camera is placed at the Fourier plane conjugate to the target plane in the object. The depth of focus is
                                          Δ            ⁢                                                  ⁢            z                    =                      λ                          2              ·                              NA                2                                                    ,                            (        5        )            where λ is the wavelength and NA is the numerical aperture. The depth of focus for the system with the transverse resolution of 9 μm is 131 μm. The volumetric targets (tumor spheroids) are typically thicker than the depth of focus. The spheroids range in size from 300 um to 1 mm. To minimize out-of-focus in the numerical reconstruction, the object plane is placed about ⅓ of the way into the tumor from the incident face. In this way, the tumor images remain in focus, except for the back face of the tumor, where multiple scattering and the “showerglass effect” already limit the imaging resolution.